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# Provable Bayesian Inference via Particle Mirror Descent

; Proceedings of the 19th International Conference on Artificial Intelligence and Statistics, PMLR 51:985-994, 2016.

#### Abstract

Bayesian methods are appealing in their flexibility in modeling complex data and ability in capturing uncertainty in parameters. However, when Bayes’ rule does not result in tractable closed-form, most approximate inference algorithms lack either scalability or rigorous guarantees. To tackle this challenge, we propose a simple yet provable algorithm, Particle Mirror Descent (PMD), to iteratively approximate the posterior density. PMD is inspired by stochastic functional mirror descent where one descends in the density space using a small batch of data points at each iteration, and by particle filtering where one uses samples to approximate a function. We prove result of the first kind that, with m particles, PMD provides a posterior density estimator that converges in terms of KL-divergence to the true posterior in rate O(1/\sqrtm). We demonstrate competitive empirical performances of PMD compared to several approximate inference algorithms in mixture models, logistic regression, sparse Gaussian processes and latent Dirichlet allocation on large scale datasets.